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# Stochastic differential equation

A stochastic differential equation (SDE) of Ito-type is written as,

\begin{eqnarray*} dX(t)&=&\mu (t,X(t))dt+\sigma(t,X(t))dW(t),\\ X(0)&=&x_0. \end{eqnarray*} The term $\mu(t,X(t))$ is called**drift coefficient** and the term $\sigma;(t,X(t))$ is called **diffusion coefficient** and W(t) is a standard Brownian motion.
An SDE is really a stochastic integral equation
\[ X(t)=x_0+\int_0^t\mu(s,X(s))ds+\int_0^t\sigma(s,X(s))dW(s).\]
The first integral $\int_0^t\mu(s,X(s))ds$ is just an ordinary Riemann Integral whereas the second integral $\int_0^t\sigma(s,X(s))dW(s)$ is a stochastic integral or Ito integral.

If the drift and diffusion coefficient satisfies

This SDE has the solution $X(t)=1/(1+W(t))$ which explodes when W(t) hits -1 for the first time. This explosion time has the same distribution as $1/G^2$ where G is a standard Gaussian random variable. This means that approximately $3/4$ of the trajectories will explode within 10 time units. An SDE which is important in financial applications is the linear SDE $dX(t)=\mu X(t)dt+\sigma X(t)dW(t),X(0)=x_0$. The solution is called a Geometric Brownian motion.

Another important related equation $dX(t)=\kappa(\theta-X(t))dt+\sigma dW(t)$, gives rise to the Ornstein-Uhlenbeck process.

\begin{eqnarray*} dX(t)&=&\mu (t,X(t))dt+\sigma(t,X(t))dW(t),\\ X(0)&=&x_0. \end{eqnarray*} The term $\mu(t,X(t))$ is called

If the drift and diffusion coefficient satisfies

- $\mu(t,X(t))$ and $\sigma(t,X(t))$ are adapted to the filtration generated by the Brownian motion W.
- $|\mu(t,x)-\mu(t,y)|\leq K|x-y|$, for $x,y\in\mathbb{R}^d, t\geq 0$

$|\sigma;(t,x)-\sigma;(t,y)|\leq K|x-y|$, for $x,y\in\mathbb{R}^d, t\geq 0$

(Lipschitz-condition), - $|\mu(t,x)|^2+|\sigma(t,x)|^2\leq C(1+|x|^2)$, for $x\in\mathbb{R}^d, t\geq 0$

(Linear growth bound),

This SDE has the solution $X(t)=1/(1+W(t))$ which explodes when W(t) hits -1 for the first time. This explosion time has the same distribution as $1/G^2$ where G is a standard Gaussian random variable. This means that approximately $3/4$ of the trajectories will explode within 10 time units. An SDE which is important in financial applications is the linear SDE $dX(t)=\mu X(t)dt+\sigma X(t)dW(t),X(0)=x_0$. The solution is called a Geometric Brownian motion.

Another important related equation $dX(t)=\kappa(\theta-X(t))dt+\sigma dW(t)$, gives rise to the Ornstein-Uhlenbeck process.

Questions: Magnus Wiktorsson

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